By Stefan Liebscher
Targeted at mathematicians having at the very least a simple familiarity with classical bifurcation thought, this monograph presents a scientific category and research of bifurcations with out parameters in dynamical structures. even supposing the equipment and ideas are in short brought, a previous wisdom of center-manifold rate reductions and normal-form calculations might help the reader to understand the presentation. Bifurcations with no parameters take place alongside manifolds of equilibria, at issues the place basic hyperbolicity of the manifold is violated. the final conception, illustrated through many functions, goals at a geometrical figuring out of the neighborhood dynamics close to the bifurcation points.
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Extra resources for Bifurcation without Parameters
0/. 7) holds. 8) is translated into a non-degeneracy condition on f . 7) imposed by the structure of one conservation law and two balance laws. 0/ D S 1 S D diag. 10) Here we have normalized the imaginary part of the Hopf eigenvalue to one. 8) onto Hopf normal form. 0/ which is equivalent to S T e3 ? 0; 0; 1/T denotes the third standard unit vector. 10). a2 ; a4 / 2 R2 . 12) we can choose a basis of R3 by: 0 1 1 v1 WD @ 0 A ; 0 0 1 0 v2 WD @ a1 A ; a3 1 v3 WD @ a2 A ; a4 v 2 ; v 3 2 . S T e 3 /?
The first condition is our structural assumption, (iii,v) are non-degeneracy assumptions which are fulfilled generically, and (ii,iv) describe our bifurcation point. Signs in (iii,v) are chosen without loss of generality, by switching signs of y and , if necessary. 0; 0; 0/ D 0 at the origin, only. 0; y. /; / Á 0 along a curve. Without loss of generality, we took this curve to be the -axis. e. under small perturbations of F respecting (i) there is a point near the origin satisfying (ii–v) for the perturbed system.
1 Transcritical bifurcation. (a) Classical bifurcation with bifurcating branch of equilibria. 4) there exists a C 2 coordinate change Q D Q . 1) to the normal form (tildes omitted) xP D x. 5) See Fig. 1a. 9) and yields a two-dimensional Jordan block of the linearization at the transcritical point. Indeed, as y is not a parameter in our setting, generically it is subject to a drift to lowest possible order in the Taylor expansion. 1 (Transcritical Bifurcation Without Parameters)  Consider a C 2 vector field with a curve of equilibria.